statement stringlengths 1 2.04k | proof stringlengths 0 6.2k | type stringclasses 20
values | symbolic_name stringlengths 1 91 | library stringclasses 18
values | filename stringclasses 208
values | imports listlengths 3 22 | deps listlengths 0 43 | docstring stringlengths 0 2.22k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
big_opL_proper f g l :
(∀ k y, l !! k = Some y → f k y ≡ g k y) →
([^o list] k ↦ y ∈ l, f k y) ≡ ([^o list] k ↦ y ∈ l, g k y). | Proof. apply big_opL_gen_proper; apply _. Qed. | Lemma | big_opL_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_gen_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_proper_2 `{!Equiv A} f g l1 l2 :
l1 ≡ l2 →
(∀ k y1 y2,
l1 !! k = Some y1 → l2 !! k = Some y2 → y1 ≡ y2 → f k y1 ≡ g k y2) →
([^o list] k ↦ y ∈ l1, f k y) ≡ ([^o list] k ↦ y ∈ l2, g k y). | Proof.
intros Hl Hf. apply big_opL_gen_proper_2; try (apply _ || done).
(* FIXME (Coq #14441) unnecessary type annotation *)
intros k. assert (l1 !! k ≡@{option A} l2 !! k) as Hlk by (by f_equiv).
destruct (l1 !! k) eqn:?, (l2 !! k) eqn:?; inversion Hlk; naive_solver.
Qed. | Lemma | big_opL_proper_2 | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"assert",
"big_opL_gen_proper_2"
] | The version [big_opL_proper_2] with [≡] for the list arguments can only be
used if there is a setoid on [A]. The version for [dist n] can be found in
[algebra.list]. We do not define this lemma as a [Proper] instance, since
[f_equiv] will then use sometimes use this one, and other times
[big_opL_proper'], depen... | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opL_ne' n :
Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (=) ==> dist n)
(big_opL o (A:=A)). | Proof. intros f f' Hf l ? <-. apply big_opL_ne; intros; apply Hf. Qed. | Instance | big_opL_ne' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL",
"big_opL_ne"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_proper' :
Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (=) ==> (≡))
(big_opL o (A:=A)). | Proof. intros f f' Hf l ? <-. apply big_opL_proper; intros; apply Hf. Qed. | Instance | big_opL_proper' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL",
"big_opL_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_consZ_l (f : Z → A → M) x l :
([^o list] k↦y ∈ x :: l, f k y) = f 0 x `o` [^o list] k↦y ∈ l, f (1 + k)%Z y. | Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed. | Lemma | big_opL_consZ_l | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_cons",
"big_opL_ext"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_consZ_r (f : Z → A → M) x l :
([^o list] k↦y ∈ x :: l, f k y) = f 0 x `o` [^o list] k↦y ∈ l, f (k + 1)%Z y. | Proof. rewrite big_opL_cons. auto using big_opL_ext with f_equal lia. Qed. | Lemma | big_opL_consZ_r | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_cons",
"big_opL_ext"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_fmap {B} (h : A → B) (f : nat → B → M) l :
([^o list] k↦y ∈ h <$> l, f k y) ≡ ([^o list] k↦y ∈ l, f k (h y)). | Proof. revert f. induction l as [|x l IH]=> f; csimpl=> //. by rewrite IH. Qed. | Lemma | big_opL_fmap | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_omap {B} (h : A → option B) (f : B → M) l :
([^o list] y ∈ omap h l, f y) ≡ ([^o list] y ∈ l, from_option f u (h y)). | Proof.
revert f. induction l as [|x l IH]=> f //; csimpl.
case_match; csimpl; by rewrite IH // left_id.
Qed. | Lemma | big_opL_omap | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_zip_seq (f : nat * A → M) n len l :
length l ≤ len →
([^o list] ky ∈ zip (seq n len) l, f ky) ≡
([^o list] k ↦ y ∈ l, f (n + k, y)). | Proof.
intros Hlen. induction l as [|x l IH] in n, len, Hlen |- *; simpl.
{ by rewrite zip_nil_r. }
destruct len as [| len]; simpl in *; first lia.
rewrite Nat.add_0_r IH; last lia.
by setoid_rewrite Nat.add_succ_r.
Qed. | Lemma | big_opL_zip_seq | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_op f g l :
([^o list] k↦x ∈ l, f k x `o` g k x)
≡ ([^o list] k↦x ∈ l, f k x) `o` ([^o list] k↦x ∈ l, g k x). | Proof.
revert f g; induction l as [|x l IH]=> f g /=; first by rewrite left_id.
by rewrite IH -!assoc (assoc _ (g _ _)) [(g _ _ `o` _)]comm -!assoc.
Qed. | Lemma | big_opL_op | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_closed (P : M → Prop) f l :
P u →
(∀ x y, P x → P y → P (x `o` y)) →
(∀ k x, l !! k = Some x → P (f k x)) →
P ([^o list] k↦x ∈ l, f k x). | Proof.
intros Hunit Hop. revert f. induction l as [|x l IH]=> f Hf /=; [done|].
apply Hop; first by auto. apply IH=> k. apply (Hf (S k)).
Qed. | Lemma | big_opL_closed | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [] | Shows that some property [P] is closed under [big_opL]. Examples of [P]
are [Persistent], [Affine], [Timeless]. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opL_zip_seqZ {A} (f : Z * A → M) n len l :
length l ≤ len →
([^o list] ky ∈ zip (seqZ n len) l, f ky) ≡
([^o list] k ↦ y ∈ l, f ((n + k)%Z, y)). | Proof.
intros Hlen. rewrite /seqZ -{1}(list_fmap_id l) -prod_map_zip.
rewrite big_opL_fmap big_opL_zip_seq; last lia.
f_equiv=> i x /=. by rewrite Z.add_comm.
Qed. | Lemma | big_opL_zip_seqZ | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_fmap",
"big_opL_zip_seq"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_bind {A B} (h : A → list B) (f : B → M) l :
([^o list] y ∈ l ≫= h, f y) ≡ ([^o list] x ∈ l, [^o list] y ∈ h x, f y). | Proof.
revert f. induction l as [|x l IH]=> f; csimpl=> //. by rewrite big_opL_app IH.
Qed. | Lemma | big_opL_bind | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_app"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_sep_zip_with {A B C} (f : A → B → C) (g1 : C → A) (g2 : C → B)
(h1 : nat → A → M) (h2 : nat → B → M) l1 l2 :
(∀ x y, g1 (f x y) = x) →
(∀ x y, g2 (f x y) = y) →
length l1 = length l2 →
([^o list] k↦xy ∈ zip_with f l1 l2, h1 k (g1 xy) `o` h2 k (g2 xy)) ≡
([^o list] k↦x ∈ l1, h1 k x) `o` ([^o list] ... | Proof.
intros Hlen Hg1 Hg2. rewrite big_opL_op.
rewrite -(big_opL_fmap g1) -(big_opL_fmap g2).
rewrite fmap_zip_with_r; [|auto with lia..].
by rewrite fmap_zip_with_l; [|auto with lia..].
Qed. | Lemma | big_opL_sep_zip_with | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_fmap",
"big_opL_op"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_sep_zip {A B} (h1 : nat → A → M) (h2 : nat → B → M) l1 l2 :
length l1 = length l2 →
([^o list] k↦xy ∈ zip l1 l2, h1 k xy.1 `o` h2 k xy.2) ≡
([^o list] k↦x ∈ l1, h1 k x) `o` ([^o list] k↦y ∈ l2, h2 k y). | Proof. by apply big_opL_sep_zip_with. Qed. | Lemma | big_opL_sep_zip | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_sep_zip_with"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_empty `{Countable K} {B} (f : K → B → M) :
([^o map] k↦x ∈ ∅, f k x) = u. | Proof. by rewrite big_opM_unseal /big_opM_def map_to_list_empty. Qed. | Lemma | big_opM_empty | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_def",
"big_opM_unseal",
"map"
] | ** Big ops over finite maps | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opM_insert `{Countable K} {B} (f : K → B → M) (m : gmap K B) i x :
m !! i = None →
([^o map] k↦y ∈ <[i:=x]> m, f k y) ≡ f i x `o` [^o map] k↦y ∈ m, f k y. | Proof. intros ?. by rewrite big_opM_unseal /big_opM_def map_to_list_insert. Qed. | Lemma | big_opM_insert | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_def",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_delete `{Countable K} {B} (f : K → B → M) (m : gmap K B) i x :
m !! i = Some x →
([^o map] k↦y ∈ m, f k y) ≡ f i x `o` [^o map] k↦y ∈ delete i m, f k y. | Proof.
intros. rewrite -big_opM_insert ?lookup_delete_eq //.
by rewrite insert_delete_id.
Qed. | Lemma | big_opM_delete | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_gen_proper_2 {B} (R : relation M) f (g : K → B → M)
m1 (m2 : gmap K B) :
subrelation (≡) R → Equivalence R →
Proper (R ==> R ==> R) o →
(∀ k,
match m1 !! k, m2 !! k with
| Some y1, Some y2 => R (f k y1) (g k y2)
| None, None => True
| _, _ => False
end) →
R ... | Proof.
intros HR ??. revert m2 f g.
induction m1 as [|k x1 m1 Hm1k IH] using map_ind=> m2 f g Hfg.
{ destruct m2 as [|k x2 m2 _ _] using map_ind.
{ rewrite !big_opM_empty. by apply HR. }
generalize (Hfg k). by rewrite lookup_empty lookup_insert_eq. }
generalize (Hfg k). rewrite lookup_insert... | Lemma | big_opM_gen_proper_2 | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_delete",
"big_opM_empty",
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_gen_proper R f g m :
Reflexive R →
Proper (R ==> R ==> R) o →
(∀ k x, m !! k = Some x → R (f k x) (g k x)) →
R ([^o map] k ↦ x ∈ m, f k x) ([^o map] k ↦ x ∈ m, g k x). | Proof.
intros ?? Hf. rewrite big_opM_unseal. apply (big_opL_gen_proper R); auto.
intros k [i x] ?%list_elem_of_lookup_2. by apply Hf, elem_of_map_to_list.
Qed. | Lemma | big_opM_gen_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_gen_proper",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_ext f g m :
(∀ k x, m !! k = Some x → f k x = g k x) →
([^o map] k ↦ x ∈ m, f k x) = ([^o map] k ↦ x ∈ m, g k x). | Proof. apply big_opM_gen_proper; apply _. Qed. | Lemma | big_opM_ext | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_gen_proper",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_ne f g m n :
(∀ k x, m !! k = Some x → f k x ≡{n}≡ g k x) →
([^o map] k ↦ x ∈ m, f k x) ≡{n}≡ ([^o map] k ↦ x ∈ m, g k x). | Proof. apply big_opM_gen_proper; apply _. Qed. | Lemma | big_opM_ne | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_gen_proper",
"map"
] | The lemmas [big_opM_ne] and [big_opM_proper] are more generic than the
instances as they also give [m !! k = Some y] in the premise. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opM_proper f g m :
(∀ k x, m !! k = Some x → f k x ≡ g k x) →
([^o map] k ↦ x ∈ m, f k x) ≡ ([^o map] k ↦ x ∈ m, g k x). | Proof. apply big_opM_gen_proper; apply _. Qed. | Lemma | big_opM_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_gen_proper",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_proper_2 `{!Equiv A} f g m1 m2 :
m1 ≡ m2 →
(∀ k y1 y2,
m1 !! k = Some y1 → m2 !! k = Some y2 → y1 ≡ y2 → f k y1 ≡ g k y2) →
([^o map] k ↦ y ∈ m1, f k y) ≡ ([^o map] k ↦ y ∈ m2, g k y). | Proof.
intros Hl Hf. apply big_opM_gen_proper_2; try (apply _ || done).
(* FIXME (Coq #14441) unnecessary type annotation *)
intros k. assert (m1 !! k ≡@{option A} m2 !! k) as Hlk by (by f_equiv).
destruct (m1 !! k) eqn:?, (m2 !! k) eqn:?; inversion Hlk; naive_solver.
Qed. | Lemma | big_opM_proper_2 | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"assert",
"big_opM_gen_proper_2",
"map"
] | The version [big_opM_proper_2] with [≡] for the map arguments can only be
used if there is a setoid on [A]. The version for [dist n] can be found in
[algebra.gmap]. We do not define this lemma as a [Proper] instance, since
[f_equiv] will then use sometimes use this one, and other times
[big_opM_proper'], depend... | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opM_ne' n :
Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (=) ==> dist n)
(big_opM o (K:=K) (A:=A)). | Proof. intros f g Hf m ? <-. apply big_opM_ne; intros; apply Hf. Qed. | Instance | big_opM_ne' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM",
"big_opM_ne"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_proper' :
Proper (pointwise_relation _ (pointwise_relation _ (≡)) ==> (=) ==> (≡))
(big_opM o (K:=K) (A:=A)). | Proof. intros f g Hf m ? <-. apply big_opM_proper; intros; apply Hf. Qed. | Instance | big_opM_proper' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM",
"big_opM_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_map_to_list f m :
([^o map] k↦x ∈ m, f k x) ≡ [^o list] xk ∈ map_to_list m, f (xk.1) (xk.2). | Proof. rewrite big_opM_unseal. apply big_opL_proper'; [|done]. by intros ? [??]. Qed. | Lemma | big_opM_map_to_list | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_proper'",
"big_opM_unseal",
"map"
] | FIXME: This lemma could be generalized from [≡] to [=], but that breaks
[setoid_rewrite] in the proof of [big_sepS_sepS]. See Coq issue #14349. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opM_list_to_map f l :
NoDup l.*1 →
([^o map] k↦x ∈ list_to_map l, f k x) ≡ [^o list] xk ∈ l, f (xk.1) (xk.2). | Proof.
intros. rewrite big_opM_map_to_list.
by apply big_opL_permutation, map_to_list_to_map.
Qed. | Lemma | big_opM_list_to_map | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_permutation",
"big_opM_map_to_list",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_singleton f i x : ([^o map] k↦y ∈ {[i:=x]}, f k y) ≡ f i x. | Proof.
rewrite -insert_empty big_opM_insert/=; last eauto using lookup_empty.
by rewrite big_opM_empty right_id.
Qed. | Lemma | big_opM_singleton | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_empty",
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_unit m : ([^o map] k↦y ∈ m, u) ≡@{M} u. | Proof.
by induction m using map_ind; rewrite /= ?big_opM_insert ?left_id // big_opM_unseal.
Qed. | Lemma | big_opM_unit | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_insert",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_fmap {B} (h : A → B) (f : K → B → M) m :
([^o map] k↦y ∈ h <$> m, f k y) ≡ ([^o map] k↦y ∈ m, f k (h y)). | Proof.
rewrite big_opM_unseal /big_opM_def map_to_list_fmap big_opL_fmap.
by apply big_opL_proper=> ? [??].
Qed. | Lemma | big_opM_fmap | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_fmap",
"big_opL_proper",
"big_opM_def",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_omap {B} (h : A → option B) (f : K → B → M) m :
([^o map] k↦y ∈ omap h m, f k y)
≡ [^o map] k↦y ∈ m, from_option (f k) u (h y). | Proof.
revert f. induction m as [|i x m Hmi IH] using map_ind=> f.
{ by rewrite omap_empty !big_opM_empty. }
assert (omap h m !! i = None) by (by rewrite lookup_omap Hmi).
destruct (h x) as [y|] eqn:Hhx.
- by rewrite omap_insert Hhx //= !big_opM_insert // IH Hhx.
- rewrite omap_insert_None // de... | Lemma | big_opM_omap | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"assert",
"big_opM_empty",
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_insert_delete `{Countable K} {B} (f : K → B → M) (m : gmap K B) i x :
([^o map] k↦y ∈ <[i:=x]> m, f k y) ≡ f i x `o` [^o map] k↦y ∈ delete i m, f k y. | Proof.
rewrite -insert_delete_eq big_opM_insert; first done. by rewrite lookup_delete_eq.
Qed. | Lemma | big_opM_insert_delete | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_insert_override (f : K → A → M) m i x x' :
m !! i = Some x → f i x ≡ f i x' →
([^o map] k↦y ∈ <[i:=x']> m, f k y) ≡ ([^o map] k↦y ∈ m, f k y). | Proof.
intros ? Hx. rewrite -insert_delete_eq big_opM_insert ?lookup_delete_eq //.
by rewrite -Hx -big_opM_delete.
Qed. | Lemma | big_opM_insert_override | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_delete",
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_fn_insert {B} (g : K → A → B → M) (f : K → B) m i (x : A) b :
m !! i = None →
([^o map] k↦y ∈ <[i:=x]> m, g k y (<[i:=b]> f k))
≡ g i x b `o` [^o map] k↦y ∈ m, g k y (f k). | Proof.
intros. rewrite big_opM_insert // fn_lookup_insert.
f_equiv; apply big_opM_proper; auto=> k y ?.
by rewrite fn_lookup_insert_ne; last set_solver.
Qed. | Lemma | big_opM_fn_insert | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_insert",
"big_opM_proper",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_fn_insert' (f : K → M) m i x P :
m !! i = None →
([^o map] k↦y ∈ <[i:=x]> m, <[i:=P]> f k) ≡ (P `o` [^o map] k↦y ∈ m, f k). | Proof. apply (big_opM_fn_insert (λ _ _, id)). Qed. | Lemma | big_opM_fn_insert' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_fn_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_filter' (φ : K * A → Prop) `{∀ kx, Decision (φ kx)} f m :
([^o map] k ↦ x ∈ filter φ m, f k x)
≡ ([^o map] k ↦ x ∈ m, if decide (φ (k, x)) then f k x else u). | Proof.
induction m as [|k v m ? IH] using map_ind.
{ by rewrite map_filter_empty !big_opM_empty. }
destruct (decide (φ (k, v))).
- rewrite map_filter_insert_True //.
assert (filter φ m !! k = None) by (apply map_lookup_filter_None; eauto).
by rewrite !big_opM_insert // decide_True // IH.
... | Lemma | big_opM_filter' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"assert",
"big_opM_empty",
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_union f m1 m2 :
m1 ##ₘ m2 →
([^o map] k↦y ∈ m1 ∪ m2, f k y)
≡ ([^o map] k↦y ∈ m1, f k y) `o` ([^o map] k↦y ∈ m2, f k y). | Proof.
intros. induction m1 as [|i x m ? IH] using map_ind.
{ by rewrite big_opM_empty !left_id. }
decompose_map_disjoint.
rewrite -insert_union_l !big_opM_insert //;
last by apply lookup_union_None.
rewrite -assoc IH //.
Qed. | Lemma | big_opM_union | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_empty",
"big_opM_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_op f g m :
([^o map] k↦x ∈ m, f k x `o` g k x)
≡ ([^o map] k↦x ∈ m, f k x) `o` ([^o map] k↦x ∈ m, g k x). | Proof.
rewrite big_opM_unseal /big_opM_def -big_opL_op. by apply big_opL_proper=> ? [??].
Qed. | Lemma | big_opM_op | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_op",
"big_opL_proper",
"big_opM_def",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_closed (P : M → Prop) f m :
Proper ((≡) ==> iff) P →
P u →
(∀ x y, P x → P y → P (x `o` y)) →
(∀ k x, m !! k = Some x → P (f k x)) →
P ([^o map] k↦x ∈ m, f k x). | Proof.
intros ?? Hop Hf. induction m as [|k x ?? IH] using map_ind.
{ by rewrite big_opM_empty. }
rewrite big_opM_insert //. apply Hop.
{ apply Hf. by rewrite lookup_insert_eq. }
apply IH=> k' x' ?. apply Hf. rewrite lookup_insert_ne; naive_solver.
Qed. | Lemma | big_opM_closed | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_empty",
"big_opM_insert",
"map"
] | Shows that some property [P] is closed under [big_opM]. Examples of [P]
are [Persistent], [Affine], [Timeless]. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opM_sep_zip_with `{Countable K} {A B C}
(f : A → B → C) (g1 : C → A) (g2 : C → B)
(h1 : K → A → M) (h2 : K → B → M) m1 m2 :
(∀ x y, g1 (f x y) = x) →
(∀ x y, g2 (f x y) = y) →
(∀ k, is_Some (m1 !! k) ↔ is_Some (m2 !! k)) →
([^o map] k↦xy ∈ map_zip_with f m1 m2, h1 k (g1 xy) `o` h2 k (g2 xy)) ≡
([^... | Proof.
intros Hdom Hg1 Hg2. rewrite big_opM_op.
rewrite -(big_opM_fmap g1) -(big_opM_fmap g2).
rewrite map_fmap_zip_with_r; [|naive_solver..].
by rewrite map_fmap_zip_with_l; [|naive_solver..].
Qed. | Lemma | big_opM_sep_zip_with | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_fmap",
"big_opM_op",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_sep_zip `{Countable K} {A B}
(h1 : K → A → M) (h2 : K → B → M) m1 m2 :
(∀ k, is_Some (m1 !! k) ↔ is_Some (m2 !! k)) →
([^o map] k↦xy ∈ map_zip m1 m2, h1 k xy.1 `o` h2 k xy.2) ≡
([^o map] k↦x ∈ m1, h1 k x) `o` ([^o map] k↦y ∈ m2, h2 k y). | Proof. intros. by apply big_opM_sep_zip_with. Qed. | Lemma | big_opM_sep_zip | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_sep_zip_with",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_kmap `{Countable K1, Countable K2} {A}
(h : K1 → K2) `{!Inj (=) (=) h} (f : K2 → A → M) (m : gmap K1 A) :
([^o map] k2 ↦ y ∈ kmap h m, f k2 y) ≡ ([^o map] k1 ↦ y ∈ m, f (h k1) y). | Proof.
rewrite big_op.big_opM_unseal /big_op.big_opM_def map_to_list_kmap big_opL_fmap.
by apply big_opL_proper=> ? [??].
Qed. | Lemma | big_opM_kmap | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_fmap",
"big_opL_proper",
"big_opM_def",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_map_seq {A} (f : nat → A → M) (start : nat) (l : list A) :
([^o map] k ↦ x ∈ map_seq start l, f k x) ≡
([^o list] i ↦ x ∈ l, f (start + i) x). | Proof.
rewrite big_op.big_opM_unseal /big_op.big_opM_def.
rewrite map_to_list_seq big_opL_zip_seq //.
Qed. | Lemma | big_opM_map_seq | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_zip_seq",
"big_opM_def",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_map_seqZ {A} (f : Z → A → M) (start : Z) (l : list A) :
([^o map] k ↦ x ∈ map_seqZ start l, f k x) ≡
([^o list] i ↦ x ∈ l, f (start + i)%Z x). | Proof.
rewrite big_op.big_opM_unseal /big_op.big_opM_def.
rewrite map_to_list_seqZ big_opL_zip_seqZ //.
Qed. | Lemma | big_opM_map_seqZ | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_zip_seqZ",
"big_opM_def",
"big_opM_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_gen_proper R f g X :
Reflexive R → Proper (R ==> R ==> R) o →
(∀ x, x ∈ X → R (f x) (g x)) →
R ([^o set] x ∈ X, f x) ([^o set] x ∈ X, g x). | Proof.
rewrite big_opS_unseal. intros ?? Hf. apply (big_opL_gen_proper R); auto.
intros k x ?%list_elem_of_lookup_2. by apply Hf, elem_of_elements.
Qed. | Lemma | big_opS_gen_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_gen_proper",
"big_opS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_ext f g X :
(∀ x, x ∈ X → f x = g x) →
([^o set] x ∈ X, f x) = ([^o set] x ∈ X, g x). | Proof. apply big_opS_gen_proper; apply _. Qed. | Lemma | big_opS_ext | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_gen_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_ne f g X n :
(∀ x, x ∈ X → f x ≡{n}≡ g x) →
([^o set] x ∈ X, f x) ≡{n}≡ ([^o set] x ∈ X, g x). | Proof. apply big_opS_gen_proper; apply _. Qed. | Lemma | big_opS_ne | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_gen_proper"
] | The lemmas [big_opS_ne] and [big_opS_proper] are more generic than the
instances as they also give [x ∈ X] in the premise. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opS_proper f g X :
(∀ x, x ∈ X → f x ≡ g x) →
([^o set] x ∈ X, f x) ≡ ([^o set] x ∈ X, g x). | Proof. apply big_opS_gen_proper; apply _. Qed. | Lemma | big_opS_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_gen_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_ne' n :
Proper (pointwise_relation _ (dist n) ==> (=) ==> dist n) (big_opS o (A:=A)). | Proof. intros f g Hf m ? <-. apply big_opS_ne; intros; apply Hf. Qed. | Instance | big_opS_ne' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS",
"big_opS_ne"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_proper' :
Proper (pointwise_relation _ (≡) ==> (=) ==> (≡)) (big_opS o (A:=A)). | Proof. intros f g Hf m ? <-. apply big_opS_proper; intros; apply Hf. Qed. | Instance | big_opS_proper' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS",
"big_opS_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_elements f X :
([^o set] x ∈ X, f x) ≡ [^o list] x ∈ elements X, f x. | Proof. by rewrite big_opS_unseal. Qed. | Lemma | big_opS_elements | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_unseal"
] | FIXME: This lemma could be generalized from [≡] to [=], but that breaks
[setoid_rewrite] in the proof of [big_sepS_sepS]. See Coq issue #14349. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opS_empty f : ([^o set] x ∈ ∅, f x) = u. | Proof. by rewrite big_opS_unseal /big_opS_def elements_empty. Qed. | Lemma | big_opS_empty | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_def",
"big_opS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_insert f X x :
x ∉ X → ([^o set] y ∈ {[ x ]} ∪ X, f y) ≡ (f x `o` [^o set] y ∈ X, f y). | Proof. intros. by rewrite !big_opS_elements elements_union_singleton. Qed. | Lemma | big_opS_insert | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_fn_insert {B} (f : A → B → M) h X x b :
x ∉ X →
([^o set] y ∈ {[ x ]} ∪ X, f y (<[x:=b]> h y))
≡ f x b `o` [^o set] y ∈ X, f y (h y). | Proof.
intros. rewrite big_opS_insert // fn_lookup_insert.
f_equiv; apply big_opS_proper; auto=> y ?.
by rewrite fn_lookup_insert_ne; last set_solver.
Qed. | Lemma | big_opS_fn_insert | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_insert",
"big_opS_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_fn_insert' f X x P :
x ∉ X → ([^o set] y ∈ {[ x ]} ∪ X, <[x:=P]> f y) ≡ (P `o` [^o set] y ∈ X, f y). | Proof. apply (big_opS_fn_insert (λ y, id)). Qed. | Lemma | big_opS_fn_insert' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_fn_insert"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_union f X Y :
X ## Y →
([^o set] y ∈ X ∪ Y, f y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ Y, f y). | Proof.
intros. induction X as [|x X ? IH] using set_ind_L.
{ by rewrite left_id_L big_opS_empty left_id. }
rewrite -assoc_L !big_opS_insert; [|set_solver..].
by rewrite -assoc IH; last set_solver.
Qed. | Lemma | big_opS_union | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_empty",
"big_opS_insert"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_delete f X x :
x ∈ X → ([^o set] y ∈ X, f y) ≡ f x `o` [^o set] y ∈ X ∖ {[ x ]}, f y. | Proof.
intros. rewrite -big_opS_insert; last set_solver.
by rewrite -union_difference_L; last set_solver.
Qed. | Lemma | big_opS_delete | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_insert"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_singleton f x : ([^o set] y ∈ {[ x ]}, f y) ≡ f x. | Proof. intros. by rewrite big_opS_elements elements_singleton /= right_id. Qed. | Lemma | big_opS_singleton | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_unit X : ([^o set] y ∈ X, u) ≡@{M} u. | Proof.
by induction X using set_ind_L; rewrite /= ?big_opS_insert ?left_id // big_opS_unseal.
Qed. | Lemma | big_opS_unit | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_insert",
"big_opS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_filter' (φ : A → Prop) `{∀ x, Decision (φ x)} f X :
([^o set] y ∈ filter φ X, f y)
≡ ([^o set] y ∈ X, if decide (φ y) then f y else u). | Proof.
induction X as [|x X ? IH] using set_ind_L.
{ by rewrite filter_empty_L !big_opS_empty. }
destruct (decide (φ x)).
- rewrite filter_union_L filter_singleton_True_L //.
rewrite !big_opS_insert //; last set_solver.
by rewrite decide_True // IH.
- rewrite filter_union_L filter_single... | Lemma | big_opS_filter' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_empty",
"big_opS_insert"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_op f g X :
([^o set] y ∈ X, f y `o` g y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ X, g y). | Proof. by rewrite !big_opS_elements -big_opL_op. Qed. | Lemma | big_opS_op | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_op",
"big_opS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_list_to_set f (l : list A) :
NoDup l →
([^o set] x ∈ list_to_set l, f x) ≡ [^o list] x ∈ l, f x. | Proof.
induction 1 as [|x l ?? IHl].
- rewrite big_opS_empty //.
- rewrite /= big_opS_union; last set_solver.
by rewrite big_opS_singleton IHl.
Qed. | Lemma | big_opS_list_to_set | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_empty",
"big_opS_singleton",
"big_opS_union"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_closed (P : M → Prop) f X :
Proper ((≡) ==> iff) P →
P u →
(∀ x y, P x → P y → P (x `o` y)) →
(∀ x, x ∈ X → P (f x)) →
P ([^o set] x ∈ X, f x). | Proof.
intros ?? Hop Hf. induction X as [|x X ? IH] using set_ind_L.
{ by rewrite big_opS_empty. }
rewrite big_opS_insert //. apply Hop.
{ apply Hf. set_solver. }
apply IH=> x' ?. apply Hf. set_solver.
Qed. | Lemma | big_opS_closed | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_empty",
"big_opS_insert"
] | Shows that some property [P] is closed under [big_opS]. Examples of [P]
are [Persistent], [Affine], [Timeless]. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opS_set_map `{Countable A, Countable B} (h : A → B) (X : gset A) (f : B → M) :
Inj (=) (=) h →
([^o set] x ∈ set_map h X, f x) ≡ ([^o set] x ∈ X, f (h x)). | Proof.
intros Hinj.
induction X as [|x X ? IH] using set_ind_L.
{ by rewrite set_map_empty !big_opS_empty. }
rewrite set_map_union_L set_map_singleton_L.
rewrite !big_opS_union; [|set_solver..].
rewrite !big_opS_singleton IH //.
Qed. | Lemma | big_opS_set_map | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_empty",
"big_opS_singleton",
"big_opS_union"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_dom `{Countable K} {A} (f : K → M) (m : gmap K A) :
([^o map] k↦_ ∈ m, f k) ≡ ([^o set] k ∈ dom m, f k). | Proof.
induction m as [|i x ?? IH] using map_ind.
{ by rewrite big_opM_unseal big_opS_unseal dom_empty_L. }
by rewrite dom_insert_L big_opM_insert // IH big_opS_insert ?not_elem_of_dom.
Qed. | Lemma | big_opM_dom | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_insert",
"big_opM_unseal",
"big_opS_insert",
"big_opS_unseal",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_set_to_map `{Countable K} {A}
(f : K → A → M) (X : gset K) (g : K → A) :
([^o map] k↦a ∈ set_to_map g X, f k a) ≡ ([^o set] k ∈ X, f k (g k)). | Proof.
induction X as [|k X ? IH] using set_ind_L.
{ by rewrite set_to_map_empty big_opM_empty big_opS_empty. }
rewrite set_to_map_insert big_opS_insert //.
rewrite big_opM_insert; last by rewrite lookup_set_to_map_None.
by rewrite IH.
Qed. | Lemma | big_opM_set_to_map | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_empty",
"big_opM_insert",
"big_opS_empty",
"big_opS_insert",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_gset_to_gmap `{Countable K} {A} (f : K → A → M) (X : gset K) c :
([^o map] k↦a ∈ gset_to_gmap c X, f k a) ≡ ([^o set] k ∈ X, f k c). | Proof. by rewrite gset_to_gmap_set_to_map big_opM_set_to_map. Qed. | Lemma | big_opM_gset_to_gmap | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_set_to_map",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_gen_proper R f g X :
Reflexive R → Proper (R ==> R ==> R) o →
(∀ x, x ∈ X → R (f x) (g x)) →
R ([^o mset] x ∈ X, f x) ([^o mset] x ∈ X, g x). | Proof.
rewrite big_opMS_unseal. intros ?? Hf. apply (big_opL_gen_proper R); auto.
intros k x ?%list_elem_of_lookup_2. by apply Hf, gmultiset_elem_of_elements.
Qed. | Lemma | big_opMS_gen_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_gen_proper",
"big_opMS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_ext f g X :
(∀ x, x ∈ X → f x = g x) →
([^o mset] x ∈ X, f x) = ([^o mset] x ∈ X, g x). | Proof. apply big_opMS_gen_proper; apply _. Qed. | Lemma | big_opMS_ext | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_gen_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_ne f g X n :
(∀ x, x ∈ X → f x ≡{n}≡ g x) →
([^o mset] x ∈ X, f x) ≡{n}≡ ([^o mset] x ∈ X, g x). | Proof. apply big_opMS_gen_proper; apply _. Qed. | Lemma | big_opMS_ne | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_gen_proper"
] | The lemmas [big_opMS_ne] and [big_opMS_proper] are more generic than the
instances as they also give [x ∈ X] in the premise. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opMS_proper f g X :
(∀ x, x ∈ X → f x ≡ g x) →
([^o mset] x ∈ X, f x) ≡ ([^o mset] x ∈ X, g x). | Proof. apply big_opMS_gen_proper; apply _. Qed. | Lemma | big_opMS_proper | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_gen_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_ne' n :
Proper (pointwise_relation _ (dist n) ==> (=) ==> dist n) (big_opMS o (A:=A)). | Proof. intros f g Hf m ? <-. apply big_opMS_ne; intros; apply Hf. Qed. | Instance | big_opMS_ne' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS",
"big_opMS_ne"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_proper' :
Proper (pointwise_relation _ (≡) ==> (=) ==> (≡)) (big_opMS o (A:=A)). | Proof. intros f g Hf m ? <-. apply big_opMS_proper; intros; apply Hf. Qed. | Instance | big_opMS_proper' | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS",
"big_opMS_proper"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_elements f X :
([^o mset] x ∈ X, f x) ≡ [^o list] x ∈ elements X, f x. | Proof. by rewrite big_opMS_unseal. Qed. | Lemma | big_opMS_elements | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_unseal"
] | FIXME: This lemma could be generalized from [≡] to [=], but that breaks
[setoid_rewrite] in the proof of [big_sepS_sepS]. See Coq issue #14349. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opMS_empty f : ([^o mset] x ∈ ∅, f x) = u. | Proof. by rewrite big_opMS_unseal /big_opMS_def gmultiset_elements_empty. Qed. | Lemma | big_opMS_empty | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_def",
"big_opMS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_disj_union f X Y :
([^o mset] y ∈ X ⊎ Y, f y) ≡ ([^o mset] y ∈ X, f y) `o` [^o mset] y ∈ Y, f y. | Proof. by rewrite big_opMS_unseal /big_opMS_def gmultiset_elements_disj_union big_opL_app. Qed. | Lemma | big_opMS_disj_union | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_app",
"big_opMS_def",
"big_opMS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_singleton f x : ([^o mset] y ∈ {[+ x +]}, f y) ≡ f x. | Proof.
intros. by rewrite big_opMS_unseal /big_opMS_def gmultiset_elements_singleton /= right_id.
Qed. | Lemma | big_opMS_singleton | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_def",
"big_opMS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_insert f X x :
([^o mset] y ∈ {[+ x +]} ⊎ X, f y) ≡ (f x `o` [^o mset] y ∈ X, f y). | Proof. intros. rewrite big_opMS_disj_union big_opMS_singleton //. Qed. | Lemma | big_opMS_insert | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_disj_union",
"big_opMS_singleton"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_delete f X x :
x ∈ X → ([^o mset] y ∈ X, f y) ≡ f x `o` [^o mset] y ∈ X ∖ {[+ x +]}, f y. | Proof.
intros. rewrite -big_opMS_singleton -big_opMS_disj_union.
by rewrite -gmultiset_disj_union_difference'.
Qed. | Lemma | big_opMS_delete | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_disj_union",
"big_opMS_singleton"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_unit X : ([^o mset] y ∈ X, u) ≡@{M} u. | Proof.
by induction X using gmultiset_ind;
rewrite /= ?big_opMS_disj_union ?big_opMS_singleton ?left_id // big_opMS_unseal.
Qed. | Lemma | big_opMS_unit | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_disj_union",
"big_opMS_singleton",
"big_opMS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_op f g X :
([^o mset] y ∈ X, f y `o` g y) ≡ ([^o mset] y ∈ X, f y) `o` ([^o mset] y ∈ X, g y). | Proof. by rewrite big_opMS_unseal /big_opMS_def -big_opL_op. Qed. | Lemma | big_opMS_op | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_op",
"big_opMS_def",
"big_opMS_unseal"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_closed (P : M → Prop) f X :
Proper ((≡) ==> iff) P →
P u →
(∀ x y, P x → P y → P (x `o` y)) →
(∀ x, x ∈ X → P (f x)) →
P ([^o mset] x ∈ X, f x). | Proof.
intros ?? Hop Hf. induction X as [|x X IH] using gmultiset_ind.
{ by rewrite big_opMS_empty. }
rewrite big_opMS_insert //. apply Hop.
{ apply Hf. set_solver. }
apply IH=> x' ?. apply Hf. set_solver.
Qed. | Lemma | big_opMS_closed | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opMS_empty",
"big_opMS_insert"
] | Shows that some property [P] is closed under [big_opMS]. Examples of [P]
are [Persistent], [Affine], [Timeless]. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opL_opL {A B} (f : nat → A → nat → B → M) (l1 : list A) (l2 : list B) :
([^o list] k1↦x1 ∈ l1, [^o list] k2↦x2 ∈ l2, f k1 x1 k2 x2) ≡
([^o list] k2↦x2 ∈ l2, [^o list] k1↦x1 ∈ l1, f k1 x1 k2 x2). | Proof.
revert f l2. induction l1 as [|x1 l1 IH]; simpl; intros Φ l2.
{ by rewrite big_opL_unit. }
by rewrite IH big_opL_op.
Qed. | Lemma | big_opL_opL | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_op",
"big_opL_unit"
] | Commuting lemmas | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
big_opL_opM {A} `{Countable K} {B}
(f : nat → A → K → B → M) (l1 : list A) (m2 : gmap K B) :
([^o list] k1↦x1 ∈ l1, [^o map] k2↦x2 ∈ m2, f k1 x1 k2 x2) ≡
([^o map] k2↦x2 ∈ m2, [^o list] k1↦x1 ∈ l1, f k1 x1 k2 x2). | Proof. repeat setoid_rewrite big_opM_map_to_list. by rewrite big_opL_opL. Qed. | Lemma | big_opL_opM | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opM_map_to_list",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_opS {A} `{Countable B}
(f : nat → A → B → M) (l1 : list A) (X2 : gset B) :
([^o list] k1↦x1 ∈ l1, [^o set] x2 ∈ X2, f k1 x1 x2) ≡
([^o set] x2 ∈ X2, [^o list] k1↦x1 ∈ l1, f k1 x1 x2). | Proof. repeat setoid_rewrite big_opS_elements. by rewrite big_opL_opL. Qed. | Lemma | big_opL_opS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opL_opMS {A} `{Countable B}
(f : nat → A → B → M) (l1 : list A) (X2 : gmultiset B) :
([^o list] k1↦x1 ∈ l1, [^o mset] x2 ∈ X2, f k1 x1 x2) ≡
([^o mset] x2 ∈ X2, [^o list] k1↦x1 ∈ l1, f k1 x1 x2). | Proof. repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL. Qed. | Lemma | big_opL_opMS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opMS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_opL {A} `{Countable K} {B}
(f : K → A → nat → B → M) (m1 : gmap K A) (l2 : list B) :
([^o map] k1↦x1 ∈ m1, [^o list] k2↦x2 ∈ l2, f k1 x1 k2 x2) ≡
([^o list] k2↦x2 ∈ l2, [^o map] k1↦x1 ∈ m1, f k1 x1 k2 x2). | Proof. symmetry. apply big_opL_opM. Qed. | Lemma | big_opM_opL | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opM",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_opM `{Countable K1} {A} `{Countable K2} {B}
(f : K1 → A → K2 → B → M) (m1 : gmap K1 A) (m2 : gmap K2 B) :
([^o map] k1↦x1 ∈ m1, [^o map] k2↦x2 ∈ m2, f k1 x1 k2 x2) ≡
([^o map] k2↦x2 ∈ m2, [^o map] k1↦x1 ∈ m1, f k1 x1 k2 x2). | Proof. repeat setoid_rewrite big_opM_map_to_list. by rewrite big_opL_opL. Qed. | Lemma | big_opM_opM | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opM_map_to_list",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_opS `{Countable K} {A} `{Countable B}
(f : K → A → B → M) (m1 : gmap K A) (X2 : gset B) :
([^o map] k1↦x1 ∈ m1, [^o set] x2 ∈ X2, f k1 x1 x2) ≡
([^o set] x2 ∈ X2, [^o map] k1↦x1 ∈ m1, f k1 x1 x2). | Proof.
repeat setoid_rewrite big_opM_map_to_list.
repeat setoid_rewrite big_opS_elements. by rewrite big_opL_opL.
Qed. | Lemma | big_opM_opS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opM_map_to_list",
"big_opS_elements",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opM_opMS `{Countable K} {A} `{Countable B} (f : K → A → B → M)
(m1 : gmap K A) (X2 : gmultiset B) :
([^o map] k1↦x1 ∈ m1, [^o mset] x2 ∈ X2, f k1 x1 x2) ≡
([^o mset] x2 ∈ X2, [^o map] k1↦x1 ∈ m1, f k1 x1 x2). | Proof.
repeat setoid_rewrite big_opM_map_to_list.
repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL.
Qed. | Lemma | big_opM_opMS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opMS_elements",
"big_opM_map_to_list",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_opL `{Countable A} {B}
(f : A → nat → B → M) (X1 : gset A) (l2 : list B) :
([^o set] x1 ∈ X1, [^o list] k2↦x2 ∈ l2, f x1 k2 x2) ≡
([^o list] k2↦x2 ∈ l2, [^o set] x1 ∈ X1, f x1 k2 x2). | Proof. symmetry. apply big_opL_opS. Qed. | Lemma | big_opS_opL | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opS"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_opM `{Countable A} `{Countable K} {B}
(f : A → K → B → M) (X1 : gset A) (m2 : gmap K B) :
([^o set] x1 ∈ X1, [^o map] k2↦x2 ∈ m2, f x1 k2 x2) ≡
([^o map] k2↦x2 ∈ m2, [^o set] x1 ∈ X1, f x1 k2 x2). | Proof. symmetry. apply big_opM_opS. Qed. | Lemma | big_opS_opM | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_opS",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_opS `{Countable A, Countable B}
(X : gset A) (Y : gset B) (f : A → B → M) :
([^o set] x ∈ X, [^o set] y ∈ Y, f x y) ≡ ([^o set] y ∈ Y, [^o set] x ∈ X, f x y). | Proof. repeat setoid_rewrite big_opS_elements. by rewrite big_opL_opL. Qed. | Lemma | big_opS_opS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opS_opMS `{Countable A, Countable B}
(X : gset A) (Y : gmultiset B) (f : A → B → M) :
([^o set] x ∈ X, [^o mset] y ∈ Y, f x y) ≡ ([^o mset] y ∈ Y, [^o set] x ∈ X, f x y). | Proof.
repeat setoid_rewrite big_opS_elements.
repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL.
Qed. | Lemma | big_opS_opMS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opMS_elements",
"big_opS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_opL `{Countable A} {B}
(f : A → nat → B → M) (X1 : gmultiset A) (l2 : list B) :
([^o mset] x1 ∈ X1, [^o list] k2↦x2 ∈ l2, f x1 k2 x2) ≡
([^o list] k2↦x2 ∈ l2, [^o mset] x1 ∈ X1, f x1 k2 x2). | Proof. symmetry. apply big_opL_opMS. Qed. | Lemma | big_opMS_opL | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opMS"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_opM `{Countable A} `{Countable K} {B} (f : A → K → B → M)
(X1 : gmultiset A) (m2 : gmap K B) :
([^o mset] x1 ∈ X1, [^o map] k2↦x2 ∈ m2, f x1 k2 x2) ≡
([^o map] k2↦x2 ∈ m2, [^o mset] x1 ∈ X1, f x1 k2 x2). | Proof. symmetry. apply big_opM_opMS. Qed. | Lemma | big_opMS_opM | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opM_opMS",
"map"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_opS `{Countable A, Countable B}
(X : gmultiset A) (Y : gset B) (f : A → B → M) :
([^o mset] x ∈ X, [^o set] y ∈ Y, f x y) ≡ ([^o set] y ∈ Y, [^o mset] x ∈ X, f x y). | Proof. symmetry. apply big_opS_opMS. Qed. | Lemma | big_opMS_opS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opS_opMS"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
big_opMS_opMS `{Countable A, Countable B}
(X : gmultiset A) (Y : gmultiset B) (f : A → B → M) :
([^o mset] x ∈ X, [^o mset] y ∈ Y, f x y) ≡ ([^o mset] y ∈ Y, [^o mset] x ∈ X, f x y). | Proof. repeat setoid_rewrite big_opMS_elements. by rewrite big_opL_opL. Qed. | Lemma | big_opMS_opMS | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [
"big_opL_opL",
"big_opMS_elements"
] | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 | |
: ∀ {A} (R : relation A), RewriteRelation R | := {}. | Instance | ∀_RewriteRelation | iris.algebra | iris/algebra/big_op.v | [
"stdpp",
"functions",
"gmap",
"gmultiset",
"iris.algebra",
"monoid",
"iris.prelude",
"options"
] | [] | The ssreflect rewrite tactic only works for relations that have a
[RewriteRelation] instance. For the purpose of this section, we want to
rewrite with arbitrary relations, so we declare any relation to be a
[RewriteRelation]. | https://gitlab.mpi-sws.org/iris/iris | 49cc21a1d9be19176b8546a5c5cb0076bf7b14e6 |
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